Lennie Bradford

N-point DFT of a sequence x(n) defined for n = 0, 1, 2 ---- N -1 is given by:

\(X\left( k \right)=\sum_{n=0}^{N-1} x\left( n \right){{e}^{-jk\omega n}}\)

\(\omega=\frac{2\pi }{N}\)

Application:

Given  x =

\(X\left( k \right)=\sum_{n=0}^{N-1} x\left( n \right){{e}^{-jk\omega n}}\)

\(X\left( 0 \right)=\sum_{n=0}^{3} x\left( n \right)\)

∴ X(0) = x(0) + x(1) +x(2) + x(3)

X(0) = 1 + 0 - 1 + 0

X(0) = 0

Similarly:

\(X\left( 1 \right)=\sum_{n=0}^{3} x\left( n \right){{e}^{-j\omega n}}\)

Putting \(\omega=\frac{2\pi }{N}\) to the above equation, we get:

\(X\left( 1 \right)=x(0)+x(1){{e}^{-j(\frac{2\pi}{4})n}}+x(2){{e}^{-j2(\frac{2\pi}{4}) n}}+x(3){{e}^{-j3(\frac{2\pi}{4}) n}}\)

X(1) = 1 + 0 (-1)(-1) + 0

X(1) = 2

Similarly:

\(X\left( 2 \right)=x(0)+x(1){{e}^{-j2(\frac{2\pi}{4})n}}+x(2){{e}^{-j4(\frac{2\pi}{4}) n}}+x(3){{e}^{-j6(\frac{2\pi}{4}) n}}\)

X(2) = 1 + 0 + (-1)(1) + 0

X(2) = 0

\(X\left( 3 \right)=x(0)+x(1){{e}^{-j6(\frac{2\pi}{4})n}}+x(2){{e}^{-j12(\frac{2\pi}{4}) n}}+x(3){{e}^{-j18(\frac{2\pi}{4}) n}}\)

Skindulgence

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THE MAHARASHTRA MUNICIPAL COUNCILS, NAGAR PANCHAYATS AND INDUSTRIAL TOWNSHIPS ACT, 1965 ————CONTENTS PREAMBLE SECTIONS PAGES CHAPTER I

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